General properties of staircase and convex dual feasible functions

Transcription

1 General properties of staircase and convex dual feasible functions JÜRGEN RIETZ, CLÁUDIO ALVES, J. M. VALÉRIO de CARVALHO Centro de Investigação Algoritmi da Universidade do Minho, Escola de Engenharia Dept. Produção e Sistemas, Universidade do Minho Braga PORTUGAL juergen_rietz@gmx.de, {claudio, vc}@dps.uminho.pt Abstract: Dual feasible functions have been used successfully to compute lower bounds and valid inequalities for different combinatorial optimization problems. In this paper, we show that some maximal dual feasible functions proposed in the literature are dominated by others under weak prerequisites. Furthermore, we explore the relation between superadditivity and convexity, and we derive new results for the case where dual feasible functions are convex. Computational results are reported to illustrate the results presented in this paper. Key-Words: Dual feasible functions; Maximal functions; Extreme functions; Dominance; Convex functions; Lower bounds. 1 Introduction Dual feasible functions were proposed originally by Johnson in [7]. Since then, different functions were defined and used to compute lower bounds and valid inequalities for general problems including cutting and packing, vehicle routing, and scheduling problems [1, 2, 3, 4, 11]. A function, -, - is called a dual feasible function (DFF), if for any finite set * + of nonnegative real numbers, the following holds Given a DFF, -, - and an instance of the one-dimensional cutting stock problem (1D-CSP) with items of lengths to be cut in the order demands from initial material of length, a valid lower bound for the optimal objective function value of the continuous relaxation (and therefore for the integer problem too) is, A DFF is a maximal dual feasible function (MDFF), if there is no other DFF, -, - with for all, -. A MDFF, -, - is extreme [8], if MDFFs, -, - with for all, - are necessarily identical with. To obtain high values for the bound, -, obviously only MDFFs should be used. Moreover, if are MDFFs with then, -, -, -, hence, - *, -, -+. Therefore extreme MDFFs should be preferred. A first characterization of MDFFs was given in [4]. A function, -, - is a MDFF, if and only if is symmetric, i.e. (1) for all [ ] (2) and the following superadditivity condition holds: for all with To prove that a function, -, - is a MDFF, the superadditivity needs to be shown for and only (see [8]). In [5], the authors surveyed the different DFFs described in the literature. In the sequel, we recall ISSN: Issue 6, Volume 8, June 2011

2 the formal definition of the functions that will be explored in this paper. The parameters will be omitted when it is possible. o, proposed by Fekete and Schepers in [6], with * +: { That is a very weak prerequiste, because usually all data are rational, hence multiplying them by the smallest common multiple of the denominators yields an equivalent instance with integer data only. In Section 3 we will discuss functions, which are convex on 0 1. It comes out that these functions easily lead to MDFFs. Section 4 contains computational tests. Conclusions and summary are given in Section 5. o o, obtained from a function proposed originally in [11], using a procedure to make it maximal [5], for any * +: { * +, proposed by Carlier, Clautiaux and Moukrim in [3], for any and : { 2 Dominance between MDFFs The characteristic mark of a MDFF, -, - is by definition, that there is no other DFF, -, - with for all, -, i.e. there is no dominance relation on the entire interval between both functions. Nevertheless, sometimes one MDFF (even if it is extreme) can be replaced by another MDFF in the calculation of the lower bound, - for the optimal objective function value of an instance of the 1D-CSP, such that, -, - is obtained. This is possible, because the arguments for the functions belong only to a discrete set. 2.1 versus o, proposed by Burdett and Johnson in [2], with and : { } This function is continuous on, - and yields for * + the identity function. Note that the non-integer part of a real value is denoted by, i.e., for. All these functions are maximal, but not always extreme [8], e.g. and are extreme for all possible parameters, while is extreme if and only if [9]. Although and are extreme, we will show in the following section that in the computation of valid lower bounds for the 1D-CSP these functions are dominated by others, i.e. they can be replaced by others without getting worse results, if all data in the given instance are integer. The following proposition shows a dominance between the Fekete and Schepers function and the one by Burdett and Johnson [2]. On the contrary, there is no such relation with respect to the function by Carlier, Clautiaux and Moukrim [3]. Moreover, we will present an example, where [ ] (with a certain parameter choice) cannot be achieved by [ ] for any possible parameter. Proposition 1. For any * +, the choice leads to. /. /. A similar relation does not exist between and for any. Proof. Both functions and are MDFFs, hence the assertion needs to be verified for only. Since * + that allows the assumption. Let with and. Since one obtains ISSN: Issue 6, Volume 8, June 2011

3 . / {. / Three cases have now to be distinguished: 1. yields. /, hence. 2 3/. /. /. / 2. : since are integers, it follows that. We get, because the nominator of the last fraction equals. /. Therefore,. /. 2 3/. 2 3/. /. 3. : we get and,,, and hence. /. /. 2 3/. 2 We get for 3/. /. from combination of the second and third case } the. /, hence. / and analogously. / because of the monotony. For the second assertion let and * +. That yields the function values 0, and for. /. Suppose there is a with. / and. /. The first equation implies, hence. The second demand yields and due to, leading to the contradiction. /. /. This proposition states mainly that results, which were calculated with, can also be obtained with. On the contrary, it will be shown in the following that sometimes yields better bounds than for any possible parameter. For any instance of the 1D-CSP with integer data only, a valid lower bound for the optimal objective function value of the integer problem arising from the Fekete and Schepersfunction can be achieved also with a Burdett and Johnson-function. Since the latter function may give better results, becomes superfluous for this purpose. To calculate the maximal possible value for [ ], according to [10], only parameters need to be tested, which belong to the set.2 * + 3 / * +. Nevertheless, trying all these possibilities is impossible, because there are still infinite many elements in this set, and in spite of restricting by a certain constant from above, the complexity would be too high. On the other hand, we can try to approximate by an element of using continued fractions, i.e. we are looking for numbers with and. The fraction is never integer, because divides the nominator, but not the denominator, hence the greatest common divider (gcd) of nominator and denominator equals, since. Therefore, the calculation can be done with the following simple algorithm: 1. Initialize the numbers,,,,,, and. Obtain the material bound [ ]. 2. If then calculate [ ( )] and update the best known lower bound if this bound is better than the current one. ISSN: Issue 6, Volume 8, June 2011

4 3. Let,, and. 4. If then go to the step 2. The numbers increase exponentially, such that the loop is left after at most repetitions. In this way, good results can be achieved with small complexity, as illustrated in Section , and the identity function Similarly to the previous subsection, the improved Vanderbeck function is dominated by, but there is no such dominance relation to. Moreover, dominates the identity function. Proposition 2. For any with and any * +, the use of leads to. /. / for all with. Additionally,. / holds for all these. On the other hand, cannot always be replaced in the same way by. Proof. Since and are MDFFs, it is enough for the first and second part to verify the proposition for. Then, it follows that. / and. /. For any, it holds that. One has. /, because. If then. /. / and and therefore. /. /. /. If then. /. /. The choice yields. /, because. For the last assertion, choose any with. Let and. Suppose, there is a with and. /. / and. /. /. The latter case yields. /, i.e., and. The case leads to, hence, implying the contradiction. Therefore, we get and, hence {. One has. /. / } { }. Since and, one gets, hence, therefore and contradiction to. /. Here the same discussion as after Proposition 1 applies. Moreover, the structure of is less complex than the one of. Computational tests to compare both functions are presented in Section 4. The following example illustrates that [ ] can be above every possible value of [ ]. Example. One piece of length 7 and one of length 15 have to be cut from initial material of length 21. That yields [ ]. /. /. This value cannot be reached by for any feasible parameter, because yields. / and of course. /, while the choice leads to. / and. /. This dominance of over can only happen if there are items of sizes in. in /. If there are only small pieces (shorter than ) then the highest values of [ ] are found with a large non-integer part. In that case set. That leads to and for. / and finally, because if then due to. Because of the negative assertions in the propositions 1 and 2, there is no dominance relation ISSN: Issue 6, Volume 8, June 2011

5 between and, hence both functions should be used to calculate valid bounds. 3 Convex functions In this section, a connection between superadditivity and convexity is developed and the resulting MDFFs are analyzed. If a real function is continuous and obeys the equation for all, then is necessarily linear. Since superadditivity allows also greater than, convex functions could be very useful for generating MDFFs under the prerequisite (1). Let be a convex set. A function is convex if for all and it holds that ( ). Theorem 1. Let be a constant. Any convex function, - fulfilling the condition (1) is superadditive in the entire domain, -. Moreover, if, - fulfills the conditions (1) and (2) and is convex on 0 1 then is a MDFF. this yields, hence is superadditive in the interval, -. To prove the second assertion, the superadditivity on the entire interval 1 1 must be shown. For that purpose assume now. Since f is symmetric, one has. /. Because is convex on 0 1, it holds that. /. / for all, hence ( ) for all. Therefore,, and, implying. Any convex function is continuous in the inner of its domain, but not necessarily differentiable, for instance. Therefore, the prerequisite, that should be convex on 0 1, is strong, but not too strong. This leads to a wide range of MDFFs. However, many of these functions will not be extreme as shown at the end of this section. Example. The following function, -, - is a MDFF: Proof. Let be given with. First it is shown, that, where describes the straight line through the points ( ) and ( ), i.e. { (3) Since is convex in the interval, -, one has, hence ( ) In the same way, can be shown. Taking yields. The straight line can be written generally in the form with certain. For Proof. fulfills the conditions (1) and (2) obviously. For one obtains and, hence is strictly convex in. /. Since is continuous, Theorem 1 can be applied. The following assertion will be used at the end of this section to simplify the proof of Proposition 4, where we show that there are only two extreme piecwise linear MDFFs, which are convex on 0 1. Proposition 3. If an extreme MDFF, -, -, different from the Fekete and Schepersfunction with, is convex in. /, then is continuous on, -. Proof. Due to the convexity in. / it follows immediately that is continuous in. /. Since ISSN: Issue 6, Volume 8, June 2011

6 for, the continuity holds also at the spot 0. Let. Clearly, due to the restriction on. If would hold, then let 2 3 and define functions, -, - as { For one has. The convexity of in. / implies in conjuction with that are convex in. / too. Moreover and are MDFFs due to Theorem 1. Therefore, would not be extreme. This contradiction proves, hence is continuous at too. A clue to disprove that a given differentiable function like (3), which is constructed from a convex function according to Theorem 1, is extreme, is the following. Draw the tangents to the graph of in the points and. /. If isn't the identity function, then both tangents have a unique common point with because of the convexity of in 0 1. Let {. /. /, - (4) Then, -, - is obviously a MDFF. If the function, -, - with for all, - is superadditive, then is not extreme. In the case of the function (3) one has and. /, hence. This approach needs not to work always. To show that, a special differentiable convex (but not strict convex) function is constructed, which can be used with Theorem 1 as counter example. First choose a very small. Let initially { Now replace in an environment of by a strict convex function, such that becomes differentiable in the whole interval. /. That yields and. /, hence. For some. / it follows that, but, where is the function (4). Hence,, yielding that the other function is not monotonous and therefore not a MDFF. Let and be metrical spaces with metrics and, respectively. A uniform continuous function has the property that for each there is a, such that for any with it holds that ( ). Any continuous function on a compact set is there uniformly continuous. If the domain of is not compact, then this property of is more than the point-by-point continuity where may depend on the point or. In the following, sometimes a derivative of a function will be discussed on a closed interval, -. In that case we mean that exists in an open set, which contains, -. Theorem 2. Let, -, - be a MDFF, such that is convex on 0 1. If there is an. /, such that is twice continuously differentiable in an environment of and if ( ), then is not extreme. Proof. Due to the continuity of in an environment of and ( ) there are numbers with and, such that has a continuous second derivative on, - and for all, -. As a MDFF, is monotonous, hence. For an enough small 1 1, let and {. / ISSN: Issue 6, Volume 8, June 2011

7 The function is once continuously differentiable, and it holds that for all, because. The functions are defined by and { ( ) hence for 0 1. Moreover, are continuously differentiable on, -, even twice in. It remains to show that are monotonously increasing on, - and. For one has and, hence has turning points at with and. Therefore,, i.e. is strictly monotonous, hence for all, -. Since for, one has for, i.e.. Because of for all it follows for that ( ) ( ). /. /, hence is monotonously increasing on, - due to the mean value theorem of differential calculus. Investigating, one has first ( ). /. Since is uniformly continuous on, -, there is a constant, such that for all, - with. If then and therefore ( )., hence. 4 / 5 / The new restriction. 2 3 guarantees for all and therefore also the monotony of. One opportunity to avoid the prerequisites of Theorem 2 is the usage of piecewise linear functions, but that does not guarantee extremality, as the following proposition shows. Proposition 4. Let, -, - be a MDFF different from and from the identity function, such that is convex in. / and piecewise linear. Then is not extreme. Proof. Suppose that is extreme. Then Proposition 3 states that is continuous on, -. We have to distinguish several cases about the intervals of different slopes. A. Let us begin with the case of many such intervals, i.e. assume that there are with and for * +, such that and for all, -, * +. The idea will be to replace in the interval by two other piecewise linear functions, such that they remain convex in 0 1. Let { Now will be shown. Indeed, we have and, hence, implying and again 0 1 by {. Define the functions 0 1, -, - } ISSN: Issue 6, Volume 8, June 2011

8 and, - D. If has a constant slope in the entire interval 0 1 and is continuous on, -, then is the identity function, but this case was disclosed in the prerequisites. Therefore, all cases were checked. {, - Both functions are convex, as shown next. They are piecewise linear and continuous. It remains to check the slopes: implies and. The definition of yields the other two needed inequalities, namely since, and implies, as desired. Since and are convex on 0 1, both functions can be used according to Theorem 1 to construct MDFFs different from. Since it holds that for all 0 1 and, the function is not extreme. To finish the proof, we need to check the situation of at most two intervals of different slopes in 0 1. B. If there are with and for * +, such that and for, - and * +, then the same proof can be used with. C. If has the same behaviour as in case (B.) except instead of, then we get, such that becomes the Burdett ad Johnson-function with parameter. As proved in [9], Proposition 1, is not extreme for, hence is not extreme in this case too. Theorem 2 and Proposition 4 imply that MDFFs, which are convex in. extreme. 4 Computational tests /, are generally not To illustrate the observation that dominates in the 1D-CSP with integer data, fast feasible lower bounds for instances of 1D-BPP (with order demands 1 for all items, which may occur repeatedly) were calculated with very low complexity as described in Subsection 2.1. Each of the following classes contained 1000 instances: 1. Container size a) all item sizes between 1 and 100 A) items B) items C) items b) like classes 1aA 1aC, but item sizes between 20 and 100 c) like classes 1aA 1aC, but item sizes between 35 and Same instances as in classes 1aA 1cC except the changed container size 3. Same instances as in classes 1aA 1cC except the changed container size 4. Same instances as in classes 1aA 1cC except the changed container size 5. Same instances as in classes 1aA 1cC except the changed container size 6. Same instances as in classes 1aA 1cC except the changed container size The only change between the classes (for the same subclass) was the container size; the item sizes and their order demands were exactly the same. The idea for this approach was that the ISSN: Issue 6, Volume 8, June 2011

9 instances of the classes 1cA 1cC can be solved easily to optimality. Increasing the container size, such that three small items may fit in one container, makes the instances harder to solve them exactly. For each instance, the material bound and the bounds [ ] and [ ] were calculated. The parameters for the MDFFs and were chosen on the base of continued fractions. The results presented in Table 1 were obtained in each class for the sum of the material bounds ( ), the sum and the minimum of the differences [ ], i.e. ( [ ] ) and { [ ] }, and the sum and the minimum of the differences [ ] [ ], i.e. ( [ ] [ ]) and { [ ] [ ]}. Class 2bB bC cA cB cC Table 1: Computational results (part I) For the set of instances that are not reported in Table 1, the sum was less than 20. Clearly, if there are DFFs and with, -, -, then multiplying the order demands by an integer above 1 increases the difference between the found bounds. Therefore, the highest values of could occur for large numbers of items only. The function clearly dominates, particularly in the classes 3cC and 3cB, while the computational effort remained very small. That confirms the stated opportunity to replace by. The comparison between and is much more impressive (see Table 2). We used the same instances again. The parameter was also chosen based on continued fractions. Let denote the sum of the differences [ ] [ ], i.e. ( [ ] [ ]). The minimum { [ ] [ ]} was always zero. For ease of compactness, only classes with are shown in Table 2. Class [ ] 1aA aA aB aB aC bA bA bB bB bC bC cA cA cA cB cB cB cC cC Table 2: Computational results (part II) Of course, the largest difference between the two calculated bounds occurred normally, when the order demands were the highest, namely in class 1cC. On the other hand, the effect of no dominance in the case of small items only (classes 4 6) could also be verified. Then always [ ] [ ] was found. 5 Conclusion In this paper, we proved that the MDFFs and are dominated by other MDFFs under the very weak prerequisite that all data are integer. Therefore, these functions can be replaced by others without getting worse results. On the other hand, none of the MDFFs and is dominated by the other, therefore both functions should be used to calculate valid bounds. Since it is impossible to try all feasible parameters for to replace, we proposed an algorithm to find useful parameters with very small complexity and tested it with success. Convex functions can be used to construct MDFFs. In this paper, we have shown that, except the identity function and, there is no ISSN: Issue 6, Volume 8, June 2011

10 piecewise linear extreme MDFF, which is convex in. /. [11] F. Vanderbeck. Exact algorithm for minimizing the number of setups in the one-dimensional cutting stock problem. Operations Research, 46(6): , Acknowledgement This work was partially supported by the Portuguese Science and Technology Foundation through the postdoctoral grant SFRH/BPD/45157/2008 for Jürgen Rietz and by the Algoritmi Research Center of the University of Minho for Cláudio Alves and José Valério de Carvalho. References: [1] C. Alves and J. M. Valério de Carvalho. A branch-and-price-and-cut algorithm for the pattern minimization problem. RAIRO Operations Research, 42: , [2] C. A. Burdett and E. L. Johnson. A subadditive approach to solve linear integer programs. Annals of Discrete Mathematics, 1: , [3] J. Carlier, F. Clautiaux and A. Moukrim. New reduction procedures and lower bounds for the two-dimensional bin-packing problem with fixed orientation. Computers and Operations Research, 34: , [4] J. Carlier and E. Néron. Computing redundant resources for the resource constrained project scheduling problem. European Journal of Operational Research, 176(3): , [5] F. Clautiaux, C. Alves and J. Valério de Carvalho. A survey of dual-feasible and superadditive functions. Annals of Operations Research, 179: , [6] S. Fekete and J. Schepers. New classes of fast lower bounds for bin packing problems. Mathematical Programming, 91:11 31, [7] D. S. Johnson. Near optimal bin packing algorithms, Dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts. [8] J. Rietz, C. Alves and J. M. Valério de Carvalho. Theoretical investigations on maximal dual feasible functions. Operations Research Letters, 38: , [9] J. Rietz, C. Alves and J. M. Valério de Carvalho. On the extremality of maximal dual feasible functions. Submitted, [10] J. Rietz, C. Alves and J. M. Valério de Carvalho. Worst case analysis of maximal dual feasible functions. Submitted, ISSN: Issue 6, Volume 8, June 2011